8 ideas
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
A reaction: Each expansion brings a limitation, but then you can expand again. |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
23218 | The brain has no responsibility for sensations, which occur in the heart [Aristotle] |
Full Idea: And of course, the brain is not responsible for any of the sensations at all. The correct view is that the seat and source of sensation is the region of the heart. | |
From: Aristotle (The Parts of Animals [c.345 BCE]), quoted by Matthew Cobb - The Idea of the Brain 1 | |
A reaction: [Need a reference] Hippocrates's assertion a century earlier made no impression on the great man. I wish he had been a little more circumspect with his own view. |
20057 | Philosophy of action studies the roles of psychological states in causing behaviour [Mele] |
Full Idea: The central task of the philosophy of action is the exploration of the roles played by a collection of psychological states in the etiology of intentional behaviour. | |
From: Alfred R. Mele (Springs of Action [1992], p.3), quoted by Rowland Stout - Action 5 'Psychological' | |
A reaction: Stout glosses this as a rather distinctive view, which endorses the causal theory of action (as opposed to citing reasons and justifications for actions). A key debate is whether the psychological inputs really do cause the outputs. |